If [tex]A=\left[\begin{array}{lll}3 & 1 & 5 \\ 2 & 6 & 4 \\ 8 & 7 & 9\end{array}\right][/tex] and [tex]B=\left[\begin{array}{lll}1 & 2 & 4 \\ 1 & 5 & 3 \\ 2 & 4 & 0\end{array}\right][/tex], find [tex]5A[/tex] and [tex]-2A+3B[/tex].
Answer (2)
5 A = 15 10 40 5 30 35 25 20 45 , − 2 A + 3 B = − 3 − 1 − 10 4 3 − 2 2 1 − 18
Explanation
Understanding the Problem We are given two matrices, A and B , and we need to compute 5 A and − 2 A + 3 B .
Calculating 5A To find 5 A , we multiply each element of matrix A by 5: 5 A = 5 × 3 2 8 1 6 7 5 4 9 = 15 10 40 5 30 35 25 20 45
Calculating -2A and 3B Next, we need to find − 2 A + 3 B . First, let's find − 2 A :
− 2 A = − 2 × 3 2 8 1 6 7 5 4 9 = − 6 − 4 − 16 − 2 − 12 − 14 − 10 − 8 − 18 Now, let's find 3 B :
3 B = 3 × 1 1 2 2 5 4 4 3 0 = 3 3 6 6 15 12 12 9 0
Calculating -2A + 3B Now, we add the matrices − 2 A and 3 B :
− 2 A + 3 B = − 6 − 4 − 16 − 2 − 12 − 14 − 10 − 8 − 18 + 3 3 6 6 15 12 12 9 0 = − 3 − 1 − 10 4 3 − 2 2 1 − 18
Final Answer Therefore, 5 A = 15 10 40 5 30 35 25 20 45 and − 2 A + 3 B = − 3 − 1 − 10 4 3 − 2 2 1 − 18 .
Examples
Matrix operations are used in various fields such as computer graphics, cryptography, and data analysis. For example, in computer graphics, matrices are used to represent transformations such as scaling, rotation, and translation of objects in 3D space. By performing matrix multiplication, we can combine multiple transformations into a single matrix, which can then be applied to the vertices of the object to render it on the screen. Similarly, in cryptography, matrices are used to encrypt and decrypt messages. The elements of the matrix can be used as keys to scramble the message, making it difficult for unauthorized parties to read it. In data analysis, matrices are used to represent data sets and perform statistical analysis. For example, we can use matrix factorization techniques to reduce the dimensionality of the data and identify patterns and relationships between variables.
We calculated 5 A and found it to be 15 10 40 5 30 35 25 20 45 and − 2 A + 3 B which equals − 3 − 1 − 10 4 3 − 2 2 1 − 18 .
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